The ultra-violet question in maximally supersymmetric field theories

Abstract

We discuss various approaches to the problem of determining which supersymmetric invariants are permitted as counterterms in maximally supersymmetric super Yang--Mills and supergravity theories in various dimensions. We review the superspace non-renormalisation theorems based on conventional, light-cone, harmonic and certain non-Lorentz covariant superspaces, and we write down explicitly the relevant invariants. While the first two types of superspace admit the possibility of one-half BPS counterterms, of the form $F^4$ and $R^4$ respectively, the last two do not. This suggests that UV divergences begin with one-quarter BPS counterterms, i.e. $d^2{F}^4$ and $d^4{R}^4$, and this is supported by an entirely different approach based on algebraic renormalisation. The algebraic formalism is discussed for non-renormalisable theories and it is shown how the allowable supersymmetric counterterms can be determined via cohomological methods. These results are in agreement with all the explicit computations that have been carried out to date. In particular, they suggest that maximal supergravity is likely to diverge at four loops in D=5 and at five loops in D=4, unless other infinity suppression mechanisms not involving supersymmetry or gauge invariance are at work.

56 pages, 1 figure, uses youngtab.sty. Contribution to the proceedings of the W.E. Heraeus Workhop "Quantum Gravity: Challenges and Perspectives", Bad Honnef, 14-16 April 2008. References and clarifying comments added

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• Minimal unitary representations from supersymmetry
• One-loop divergences in the 6D, N=(1,0) abelian gauge theory
• The Ultraviolet Finiteness of N=8 Supergravity
• On UV Finiteness of the Four Loop N=8 Supergravity
• On the two-loop divergences of the 2-point hypermultiplet supergraphs for $6D$, $\mathcal{N}=(1,1)$ SYM theory
• Off-shell invariants of linearized $4D,\mathcal{N}=2$ supergravity in the harmonic approach
• On the two-loop divergences in 6D, $\mathcal{N}=(1,1)$ SYM theory
• Maximally Supersymmetric Yang-Mills in five dimensions in light-cone superspace
• One-loop divergences in the six-dimensional $\mathcal{N}=(1,0)$ hypermultiplet self-coupling model
• Deformation of d=4, N> 4 Supergravities Breaks Nonlinear Local Supersymmetry
• On two-loop divergences of effective action in $6D$, $\mathcal{N}=(1,1)$ SYM theory
• Higher-dimensional invariants in $6D$ super Yang-Mills theory